1 Introduction
Abstract

We present a rigorous theory of a unified and simple inverse scattering transform (IST) for both focusing and defocusing real nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity. The IST problems for the nonlocal equations with NZBCs are more complicated then ones for the local equations with NZBCs. The suitable uniformization variable is introduced in order to make the direct and inverse problems be established on a complex plane instead of a two-sheeted Riemann surface. The direct scattering problem establishes the analyticity, symmetries, and asymptotic behaviors of Jost solutions and scattering matrix, and properties of discrete spectra. The inverse problem is formulated and solved by means of a matrix-valued Riemann-Hilbert problem. The reconstruction formula, trace formulae, and theta conditions are obtained. Finally, the dynamical behaviors of solitons for four different cases for the reflectionless potentials for both focusing and defocusing nonlocal mKdV equations with NZBCs are analyzed in detail.

Keywords: nonlocal mKdV equation; non-zero boundary conditions; Riemann surface; inverse scattering transform; matrix Riemann-Hilbert problem; solitons, breathers

A unified inverse scattering transform for the nonlocal modified KdV equation with non-zero boundary conditions

[0.2in]

Guoqiang Zhang and Zhenya Yan Email address: zyyan@mmrc.iss.ac.cn

[0.03in] Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

(Date:   July 30, 2019)

1 Introduction

Generally speaking, it is so difficult to solve the initial-value problems exactly of nonlinear partial differential equations (PDEs), but, in 1967, Gardner, Greene, Kruskal and Miura (GGKM) discovered a novel inverse scattering transform (IST) to exactly solve the initial-value problem for the important nonlinear PDEs, the Korteweg-de Vries (KdV) equation [1]. After that, numerous attempts were made to extend the application of this method in other integrable nonlinear PDEs admitting the so-called Lax pairs [2]. In 1972, Zakharov and Shabat derived the IST for the nonlinear Schrödinger (NLS) equation in terms of the Lax pair [3]. After that, Ablowitz, Kaup, Newell and Segur (AKNS) developed a class of integrable systems, called AKNS systems, and presented a general framework for the IST [4, 5]. Subsequently, many integrable nonlinear evolution equations were solved by the IST, such as the modified KdV equation [6, 7], the sine-Gordon equation [8], the Kadomtsev-Petviashvili equation [9], the Camassa-Holm equation [10], the Benjamin-Ono Equation [11] and the Degasperis-Procesi equation [12].

Since the symmetry was introduced in the generalized Hamiltonians in 1998 by Bender et al [13] and in the nonlinear Schödinger equation[14], it has been verified to play a more and more important role in many fields (see, e.g., [15, 16]). In 2013, the symmetry was introduced to the first one of the well-known AKNS system to present a nonlocal NLS equation, and outlined the IST with zero-boundary conditions (ZBCs) [17]. The nonlocal NLS equation was shown to be gauge equivalent to the unconventional system of coupled Landau-Lifshitz equations [18]. The nonlocal integrable systems are of importance significance in the theoretical study of mathematical physics and applications in the fields of nonlinear science [19]. Moreover, the multi-component local and nonlocal generalized NLS equations were recently introduced with the aid of two families of parameters[20, 21, 22]. Some other nonlocal nonlinear wave equations were presented (see, e.g., Refs. [23, 24, 25, 27, 26, 29, 28]). Some reverse space-time and inverse time integrable nonlocal nonlinear equations were introduced and their ISTs with ZBCs were presented [30, 31]. Recently, Ablowitz et al. developed the IST with non-zero boundary conditions (NZBCs) for the nonlocal NLS equation, nonlocal reverse space-time NLS equation and nonlocal reverse space-time Sine-Gordon/Sinh-Gordon equation [32, 33, 34]. In contrast to the original method for the integrable systems with NZBCs developed by Zakharov [35] using a two-sheeted Riemann surface, Ablowitz et al. introduced a uniformization variable [36] to solve the inverse problem on a standard complex -plane. This manner was also used to study the IST for the NLS equation with NZBCs by Ablowitz, Biondini, Demontis, Prinary et al. [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Recently, we developed the approach to present a rigorous theory of the IST for the focusing and defocusing KdV equations with NZBCs at infinity [53].

Recently, a new integrable real nonlocal (reverse-space-time) mKdV equation, as a -symmetric nonlocal extension of the physically impotant mKdV equation, was introduced [30, 31]

(1)

which was shown to appear in the atmospheric and oceanic dynamical system [54]. The Darboux transformation and soliton solutions of the focusing Eq. (1) with were studied [55]. Moreover, the IST for the focusing Eq. (1) with and ZBCs was presented [56]. Particularly, i) when , the nonlocal mKdV equation reduces to the usual mKdV equation; ii) when , the focusing (defocusing) nonlocal mKdV equation reduces to the defocusing (focusing) usual mKdV equation; iii) when , the focusing (defocusing) nonlocal mKdV equation reduces to the defocusing (focusing) nonlocal mKdV equation.

To the best of our knowledge, the IST for the nonlocal mKdV equation with NZBCs was not reported before. Moreover, the IST for the nonlocal mKdV equation with NZBCs is more complicated than one for the nonlocal mKdV equation with ZBCs [56]. In this paper, we will focus on the IST for the initial-value problem for both focusing and defocusing nonlocal mKdV equations (1) with NZBCs at infinity

(2)

where with . In contrast to four cases used to deal with the nonlocal NLS equation with NZBCs [32], that is, two different signs of nonlinearity and two different values of the phase difference between plus and minus infinities, we will present a uniform and simple IST to explore nonlocal integrable systems with NZBCs. For example, we here investigate the unified IST for the nonlocal mKdV equation with NZBCs.

The rest of this paper is organized as follows. In Sec. 2, the direct scattering theory for the nonlocal mKdV equation with NZBCs is deduced. By introducing a uniformization variable, the direct scattering problem can be studied in a standard complex -plane. Then the analytical domains of the Jost solutions and scattering coefficients are found. Based on the reduction conditions of the Lax pair, the basic symmetries of the modified Jost solutions and scattering matrix are also established, which generate the discrete spectrum and residue conditions. Moreover, the asymptotic behaviors for the modified Jost solutions and scattering matrix are also discovered. Sec. 3 focuses on the inverse problem with NZBCs. A generalized and uniform matrix-value Riemann-Hilbert problem (RHP) is formulated and can be solved via the Cauchy projectors and the Plemelj’s formulae. The trace formula, theta condition, and reflectionless potentials for both focusing and defocusing nonlocal mKdV equations with NZBCs are obtained. Particularly, some solutions and breather, and their interactions are illustrated. Finally, we give some conclusions and discussions in Sec.4.

2 Direct scattering problem with NZBCs

In the direct scattering theory, we will deduce the analytical properties, symmetries and asymptotic behaviors of the (modified) Jost solutions and scattering datas, and the discrete spectrum.

2.1 Lax pair and uniformization variable

The nonlocal mKdV equation (1) possesses the following Lax pair [30]

(3)
(4)

where is an iso-spectral parameter, the eigenfunction is chosen as a matrix, the potential matrix is given by

(5)

and is one of the Pauli matrices, which are

It is easy to check that the zero curvature equation just leads to Eq. (1). The only difference between the Lax pairs of nonlocal and local mKdV equations is that there are both the local function and nonlocal function in the nonlocal mKdV equation, which leads to their distinct wave structures and other properties.

Considering the asymptotic scattering problem () of the Lax pair (3, 4):

(6)

with

(7)

the fundamental matrix solution of Eq. (6) can be written as

(8)

where

(9)

A two-sheeted Riemann surface should be introduced such that is a single-valued function of on this surface. We clarify the two-sheeted Riemann surface in two cases corresponding to and . Let and . Then , , respectively, on Sheets I and II.

  • For , the branch points are . Let , then the branch cut (the discontinuity of ) is the segment . The Riemann surface is then given by gluing the two copies of the complex plane along the branch cut.

  • For , the branch points are . Let , then the branch cut (the discontinuity of ) is the segment . The Riemann surface is then given by gluing the two copies of the complex plane along the branch cut.

Figure 1: The complex -plane for (left) and (right) showing the discrete spectrums [zeros of scattering data (blue) in grey region and those of scattering data (red) in white region], the regions and for (grey) and (white), respectively, and the orientation of the contours for the Riemann-Hilbert problem.

To seek for the analytical regions of the Jost solutions and scattering datas, we usually need to determine the regions where Im . From the definition of the two-sheeted Riemann surface, one obtains that the region where Im is the upper-half plane (UHP) on Sheet-I and the lower-half plane (LHP) on Sheet-II and the region where Im is the LHP on Sheet I and UHP on Sheet II.

The IST with NZBCs is first presented by Zakharov in 1973 to solve the initial-value problem for the NLS equation [35], where the two-sheeted Riemann surface was employed. An improvement was made with the introduction of a uniformization variable [36], which transformed the scattering problem onto a standard complex -plane. Define the uniformization variable by the conformal mapping as

(10)

and the inverse mapping is deduced by

(11)

One observes the mapping relation between the two-sheeted Riemann -surface and complex -plane in two cases corresponding to and :

  • As , the mapping relation is observed as follows (see Fig. 1(left)):

    • The Sheet-I and Sheet-II are mapped, respectively, onto the exterior and interior of the circle of radius ;

    • The branch cut are mapped onto the circle of radius ;

    • is mapped onto the real axis;

    • The region where Im (Im ) of the Riemann surface is mapped onto the grey (white) domain in the complex -plane.

  • As , the mapping relation is observed as follows (see Fig. 1(right)):

    • The Sheet-I and Sheet-II are mapped, respectively, onto the exterior and interior of the circle of radius ;

    • The branch cut is mapped into the circle of radius ;

    • The real axis is mapped onto the real axis;

    • The region for Im (Im ) of the Riemann surface is mapped onto the grey (white) domain in the complex -plane.

For convenience, we define

(12)
(13)

which can generate for and for .

2.2 Jost solutions and modified forms

Since, as usual, the continuous spectrum of is the set of all values of satisfying [42], then we denote the continuous spectrum by . Then it follows from the above-mentioned analysis that we have

(14)

The Jost solutions are the simultaneous solutions of the Lax pair (3, 4) satisfying

(15)

For convenience, we introduce the modified Jost solutions by factorizing the asymptotic exponential oscillations:

(16)

such that

(17)

By the constant variation approach from Eq. (3), one can write the modified Jost solutions as

(18)

where , and . Next, we will establish the existence, uniqueness, continuity and analyticity of the Jost solutions. For convenience, we let as and as .

Lemma 1.

Given a series and a function on the interval , where and are matrix-valued functions. Suppose converges uniformly on the interval and , , then and

The lemma is proved easily, which is auxiliary to confirm the existence of the Jost solutions.

Proposition 1.

Suppose , then the Jost integral equation (18) has unique solutions defined by (16) in . Moreover, the columns and can be extended analytically to and continuously to , and and can be extended analytically to and continuously to , where and are referred to the first and second columns of , respectively.

Proof.

We prove the proposition in two cases corresponding to and .

As , we let , which satisfy from (18) that

In the following, we put as a example to claim the proof. The proofs of other column and are similar. satisfies that

(19)

where . Introduce the Neumann series for as

(20)

with ’s defined by

For and , we have

For , let

(21)

where . As one restricts to the domain , there exists a constant such that for , which gives the following recursion inequality:

(22)

One can establish the estimate for each term of the series by induction as

(23)

which implies that the series converges uniformly in the domain . Thus is continuous in and analytic in corresponding to and corresponding to . Besides, the arbitrariness of infers that is continuous in and analytic in . As , Eq. (23) infers that

(24)

which derives that converges uniformly in . In terms of Lemma 1, one obtains that as , one has

(25)

By the arbitrariness of , the existence of the solution for Eq. (19) follows. It follows from the following inequality

(26)

that the asymptotic of is given as

To prove the uniqueness of the solution for integral equation (19), we only need to prove that the homogeneous integral equation has only zero solution. The homogeneous integral equation reads

It follows easily that

(27)

For convenience, let

(28)

Then we have

which leads to

that is, . Therefore, follows from in Eq. (27). This completes the proof of the uniqueness. Similarly, the existence, uniqueness, continuity and analyticity of can also follow trivally from .

As , solves the following integral equation:

where is the argument of . In the same manner, we introduce a Neumann series

with

For , one has

We find the recursion inequality:

As , the following inequalities hold:

Then one can determine the estimate for each term of the series as

where

Thus converges.

The inequality

discoveries the asymptotic of as

In the same approach, one only needs to prove that the homogeneous integral equation has only zero solution for uniqueness, which reads

For convenience, let

(29)

then one has

The monotonicity leads to and further . The continuity in Eq. (18):

derives that is continuous in . Thus, the proof follows. ∎

Corollary 1.

Suppose , then Eq. (3) has unique solutions defined by (15) in . Besides, and can be extended analytically to and continuously to while and can be extended analytically to and continuously to .

Proof.

The proof follows from Proposition 1 by Eq. (16). ∎

Lemma 2 (Liouville’s formula).

Consider an -dimensional first-order homogeneous linear differential equation on an interval of the real line, where for denotes a square matrix of dimension with real or complex entries. Let denote a matrix-valued solution on . If the trace is a continuous function, then one has

Proposition 2.

The Jost solutions are the simultaneous solutions of both parts of the Lax pair (3, 4).

Proof.

We only need to show that solve the -part (4). It follows from the compatibility condition of the Lax pair (3, 4), , that we have

that is, solve the -part (3). Liouville’s formula implies that

(30)

Thus are the fundamental matrix solutions as . There exist two matrices such that